calculate the time elapsed for a robot to pass certain distance with a load [closed]

For a robotics project I wanted to find the optimal gear ratio for my robot to travel 10 meters. Unfortunately. the acceleration is nonconstant, and that proved to make my life much harder. I think I have most of it figured out, and the only part left is to find the correlation between the acceleration and velocity graph to the acceleration/velocity/position to time graph.

The robot with mass m and wheels that have a radius of r is allowed to accelerate on a surface with negligible friction. It uses 4 motors that have a free speed of Vfree rpm, and a stall torque of Tstall. The graph is included below. How can I find the time taken for this robot to travel x meters? I was able to turn the torque to angular velocity graph into an acceleration to velocity graph, but I got stuck trying to transform it into a distance to time or even a velocity to time graph.

closed as off-topic by John Rennie, glS, JMac, ZeroTheHero, AccidentalFourierTransformAug 30 '18 at 15:38

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• If that helps, you may use $a=v\frac{dv}{ds}$ to find displacement. $a$ is acceleration, $v$ is velocity and $s$ is displacement. If displacement does not change direction, then it can be taken as distance too. – Jitendra Aug 26 '18 at 11:07
• If you don't get an answer here in a few days, you may want to try over at Robotics where perhaps someone there has dealt with a similar issue. – Kyle Kanos Aug 26 '18 at 11:58

Then write Newton’s 2nd law as $$F(v)=m\;dv/dt$$ and solve the differential equation for $$v(t)$$. Then integrate velocity to get distance.